a right angle triangle having 3 sides has area of 210 cm and perimeter is equal to 84 cm find the sum of its base and perpendicular

Question

About math's

 

 

 

 

 

Answer 1

area = ½•ab where legs are a & b

210 = ½•ab 

ab = 420 

 

perimeter = a+b+c = 84 where c is hypotenuse

so a+b = 84-c

 

(a+b)² = a² + 2ab + b² but c² = a² + b²

so (a+b)²  = c² + 2(420)

 

(84-c)² = c² + 840  after substituting for (a+b).

84² - 168c + c²  = c² + 840  and the c squared terms can be cancelled, so

84²            - 840 = 168c;  c = 37.

 

Therefore a+b = 84 - 37 = 47.  END OF SOLUTION.

 

Answer 2

let the sides be a,b, c, c being the hypotenuse and b the height

Then area =1/2(ab)=210; b=420/a

The Perimeter=a+b+c=84;b+c=84-a;a=(84-(b+c))

by pythagorean theorem a^2+b^2=c^2=(84-(a+b))^2

a^2+b^2=84^2-2(84*(a+b))+(a+b)^2

a^2+b^2=84^2-2(84*(a+b))+a^2+b^2+2ab

0=84^2-2(84*(a+b))+2ab but ab =420 and a+b=84-c

0=7056-2*84*(84-c)+2*420

0=7056-14112+168c+840

0=7056-14112+840 +168c

168c=6216

c=37

a+b=47;substututing for a from the area a=420/b

420/b+b=47

420+b^2-47b=0

(b-35)(b-12)=0; b=35 or 12.

The sides of the Triangle are a=12, b=35, and c=37

 

 

Check: Perimeter=12+35+37=84 

                   Area=1/2X12X35=210

 

Answer 3

Area = 1/2 * Base * Height

 

A = (1/2)Bh

 

Doubling the triangle creates a rectangle with twice the perimeter given.

So the perimeter of the rectangle is 2B + 2h = 2*p, where p is the perimeter of the triangle.

 

                                                     B + H = p  <--- divides everything by 2.

 

 

Plugging in:

   210 = (1/2)Bh --->   420 = Bh

 

   B + h = 84 ---> h = 84 - B

 

420 = B( 84 - B)

 

420 = 84B - B^2

 

B^2 - 84B - 420 = 0

 

B =  ( 84 +or-  sqrt( 84*84 + 4*420))/2

 

   

 = ( 84 +or- 16*sqrt(21))/2